a)\(\left(2x+3\right)\left(3y+5\right)=17\)

b) \(\left(2y+9\right)\left(11-2x\right)=57\)

c) \(\left(3x-5\right)\left(3y-2\right)=31\)

 Lần lượt xét từng trường hợp cho mỗi câu .

\(M=2x^2+9y^2-6xy-6x-12y+2028\\ =3\left(x^2-2xy+y^2\right)-\left(x^2+6x+9\right)+6\left(y^2-2y+1\right)+2025\\ =\left(x-y\right)^2-\left(x-3\right)^2+6\left(y-1\right)^2+2025\ge2025\)

Dấu \("="\Leftrightarrow\left\{{}\begin{matrix}x=y\\x=3\\y=1\end{matrix}\right.\) (vô lí) nên dấu \("="\) ko thể xảy ra

\(N=x^2-4xy+5y^2+10x-22y+28\\ =\left(x^2+4y^2+25-4xy-20y+10x\right)+\left(y^2-2y+1\right)+2\\=\left(x-2y+5\right)^2+\left(y-1\right)^2+2\ge2\)

Dấu \("="\Leftrightarrow\left\{{}\begin{matrix}x-2y=5\\y=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=7\\y=1\end{matrix}\right.\)

\(M=2x^2+9y^2-6xy-6x-12y+2028=\left(x+2\right)^2-6y\left(x+2\right)+9y^2+\left(x-5\right)^2+1999=\left(x+2-3y\right)^2+\left(x-5\right)^2+2019\ge1999\)

\(ĐTXR\Leftrightarrow\left\{{}\begin{matrix}x=5\\y=\dfrac{7}{3}\end{matrix}\right.\)

\(N=x^2-4xy+5y^2+10x-22y+28=\left(x+5\right)^2-4y\left(x+5\right)+4y^2+\left(y-1\right)^2+2=\left(x+5-2y\right)^2+\left(y-1\right)^2+2\ge2\)

\(ĐTXR\Leftrightarrow\left\{{}\begin{matrix}x=-3\\y=1\end{matrix}\right.\)

H=\(x^6-2x^3+x^2-2x+2\)

\(=x^6+2x^5+3x^4+2x^2-2x^5-4x^4-6x^3-4x^2-4x+x^4+2x^3+3x^2+2x+2\)

\(=x^2\left(x^4+2x^3+3x^2+2\right)-2x\left(x^4+2x^3+3x^2+2\right)+\left(x^4+2x^3+3x^2+2\right)\)

\(=\left(x^2-2x+1\right)\left(x^4+2x^3+3x^2+2\right)\)

\(=\left(x-1\right)^2\left(x^2+1\right)\left(x^2+2x+2\right)\)

\(=\left(x-1\right)^2\left(x^2+1\right)\left[\left(x+1\right)^2+1\right]\text{≥}0\)

Vì \(\left\{{}\begin{matrix}\left(x-1\right)^2\text{≥}0\\\left(x^2+1\right)\text{≥}1\\\left(x+1\right)^2+1\text{≥}1\end{matrix}\right.\)

⇒ MinH=0 ⇔ \(x=1\)

Lời giải:

$3xy+2x-5y=6$

$x(3y+2)-5y=6$

$3x(3y+2)-15y=18$

$3x(3y+2)-5(3y+2)=8$

$(3y+2)(3x-5)=8$

Đến đây lập bảng xét giá trị thôi bạn.

6xy+10x+9y=2

<=>2x(3y+5)+9y+15-17=0

<=>2x(3y+5)+3(3y+5)=17

<=>(2x+3)(3y+5)=17

tới đây bạn lập bảng là xong

Ta có: \(6xy+10x+9y-2=0\Leftrightarrow2x\left(3y+5\right)+9y+15-17=0\)

\(\Leftrightarrow2x\left(3y+5\right)+3\left(3y+5\right)=17\Leftrightarrow\left(2x+3\right)\left(3y+5\right)=17\)

Ta có bảng sau:

Vậy không tồn tại x, y nguyên dương thỏa mãn bài toán. 

\(3xy+2x-5y=6\)

\(\Leftrightarrow9xy+6x-15y=18\)

\(\Leftrightarrow\left(9xy+6x\right)-\left(15y+10\right)=8\)

\(\Leftrightarrow3x.\left(3y+2\right)-5\left(3y+2\right)=8\)

\(\Leftrightarrow\left(3x-5\right)\left(3y+2\right)=8\)

Do x,y nguyên nên ta có bảng sau

Bạn tự KL nhé

$3xy+2x-5y=6$

$\iff 9xy+6x-15y=18$

$\iff \left(9xy+6x\right)-\left(15y+10\right)=8$

$\iff 3x.\left(3y+2\right)-5\left(3y+2\right)=8$

$\iff \left(3x-5\right)\left(3y+2\right)=8$